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In this dissertation the electronic and high-energy optical properties of thin nanoscale
films of the magnetic topological insulator (MTI) (V,Cr)y(BixSb1-x)2-yTe3 are studied
by means of X-ray photoelectron spectroscopy (XPS) and electron energy-loss
spectroscopy (EELS). Magnetic topological insulators are presently of broad interest
as the combination of ferromagnetism and spin-orbit coupling in these materials
leads to a new topological phase, the quantum anomalous Hall state (QAHS), with
dissipation less conduction channels. Determining and controlling the physical
properties of these complex materials is therefore desirable for a fundamental understanding
of the QAHS and for their possible application in spintronics. EELS can
directly probe the electron energy-loss function of a material from which one can
obtain the complex dynamic dielectric function by means of the Kramers-Kronig
transformation and the Drude-Lindhard model of plasmon oscillations.
The XPS core-level spectra in (V,Cr)y(BixSb1-x)2-yTe3 are analyzed in detail with
regards to inelastic background contributions. It is shown that the spectra can be
accurately described based on the electron energy-loss function obtained from an
independent EELS measurement. This allows for a comprehensive and quantitative
analysis of the XPS data, which will facilitate future core-level spectroscopy studies
in this class of topological materials. From the EELS data, furthermore, the bulk and
surface optical properties were estimated, and compared to ab initio calculations
based on density functional theory (DFT) performed in the GW approximation
for Sb2Te3. The experimental results show a good agreement with the calculated
complex dielectric function and the calculated energy-loss function. The positions of
the main plasmon modes reported here are expected to be generally similar in other
materials in this class of nanoscale TI films. Hence, the present work introduces
EELS as a powerful method to access the high-energy optical properties of TI
thin films. Based on the presented results it will be interesting to explore more
systematically the effects of stoichiometry, magnetic doping, film thickness and
surface morphology on the electron-loss function, potentially leading to a better
understanding of the complex interplay of structural, electronic, magnetic and
optical properties in MTI nanostructures.
In the present thesis the MBE growth and sample characterization of HgTe structures is investigated
and discussed. Due to the first experimental discovery of the quantum Spin Hall effect
(QSHE) in HgTe quantum wells, this material system attains a huge interest in the spintronics
society. Because of the long history of growing Hg-based heterostructures here at the Experimentelle
Physik III in Würzburg, there are very good requirements to analyze this material
system more precisely and in new directions. Since in former days only doped HgTe quantum
wells were grown, this thesis deals with the MBE growth in the (001) direction of undoped
HgTe quantum wells, surface located quantum wells and three dimensional bulk layers. All
Hg-based layers were grown on CdTe substrates which generate strain in the layer stack and
provide therefore new physical effects. In the same time, the (001) CdTe growth was investigated
on n-doped (001) GaAs:Si because the Japanese supplier of CdTe substrates had a
supply bottleneck due to the Tohoku earthquake and its aftermath in 2011.
After a short introduction of the material system, the experimental techniques were demonstrated
and explained explicitly. After that, the experimental part of this thesis is displayed.
So, the investigation of the (001) CdTe growth on (001) GaAs:Si is discussed in chapter 4.
Firstly, the surface preparation of GaAs:Si by oxide desorption is explored and analyzed.
Here, rapid thermal desorption of the GaAs oxide with following cool down in Zn atmosphere
provides the best results for the CdTe due to small holes at the surface, while e.g. an atomic
flat GaAs buffer deteriorates the CdTe growth quality. The following ZnTe layer supplies the
(001) growth direction of the CdTe and exhibits best end results of the CdTe for 30 seconds
growth time at a flux ratio of Zn/Te ~ 1/1.2. Without this ZnTe layer, CdTe will grow in the
(111) direction. However, the main investigation is here the optimization of the MBE growth
of CdTe. The substrate temperature, Cd/Te flux ratio and the growth time has to be adjusted
systematically. Therefore, a complex growth process is developed and established. This optimized
CdTe growth process results in a RMS roughness of around 2.5 nm and a FWHM value
of the HRXRD w-scan of 150 arcsec. Compared to the literature, there is no lower FWHM
value traceable for this growth direction. Furthermore, etch pit density measurements show
that the surface crystallinity is matchable with the commercial CdTe substrates (around 1x10^4
cm^(-2)). However, this whole process is not completely perfect and offers still room for improvements.
The growth of undoped HgTe quantum wells was also a new direction in research in contrast
to the previous n-doped grown HgTe quantum wells. Here in chapter 5, the goal of very low
carrier densities was achieved and therefore it is now possible to do transport experiments in
the n - and p - region by tuning the gate voltage. To achieve this high sample quality, very precise
growth of symmetric HgTe QWs and their HRXRD characterization is examined. Here,
the quantum well thickness can now determined accurate to under 0.3 nm. Furthermore, the transport analysis of different quantum well thicknesses shows that the carrier density and
mobility increase with rising HgTe layer thickness. However, it is found out that the band
gap of the HgTe QW closes indirectly at a thickness of 11.6 nm. This is caused by the tensile
strained growth on CdTe substrates. Moreover, surface quantum wells are studied. These
quantum wells exhibit no or a very thin HgCdTe cap. Though, oxidization and contamination
of the surface reduces here the carrier mobility immensely and a HgCdTe layer of around 5 nm
provides the pleasing results for transport experiments with superconductors connected to the
topological insulator [119]. A completely new achievement is the realization of MBE growth
of HgTe quantum wells on CdTe/GaAs:Si substrates. This is attended by the optimization of
the CdTe growth on GaAs:Si. It exposes that HgTe quantum wells grown in-situ on optimized
CdTe/GaAs:Si show very nice transport data with clear Hall plateaus, SdH oscillations, low
carrier densities and carrier mobilities up to 500 000 cm^2/Vs. Furthermore, a new oxide etching
process is developed and analyzed which should serve as an alternative to the standard
HCl process which generates volcano defects at some time. However, during the testing time
the result does not differ in Nomarski, HRXRD, AFM and transport measurements. Here,
long-time tests or etching and mounting in nitrogen atmosphere may provide new elaborate
results.
The main focus of this thesis is on the MBE growth and standard characterization of HgTe bulk
layers and is discussed in chapter 6. Due to the tensile strained growth on lattice mismatched
CdTe, HgTe bulk opens up a band gap of around 22 meV at the G-point and exhibits therefore
its topological surface states. The analysis of surface condition, roughness, crystalline quality,
carrier density and mobility via Nomarski, AFM, XPS, HRXRD and transport measurements
is therefore included in this work. Layer thickness dependence of carrier density and mobility
is identified for bulk layer grown directly on CdTe substrates. So, there is no clear correlation
visible between HgTe layer thickness and carrier density or mobility. So, the carrier density is
almost constant around 1x10^11 cm^(-2) at 0 V gate voltage. The carrier mobility of these bulk
samples however scatters between 5 000 and 60 000 cm^2/Vs almost randomly. Further experiments
should be made for a clearer understanding and therefore the avoidance of unusable
bad samples.But, other topological insulator materials show much higher carrier densities and
lower mobility values. For example, Bi2Se3 exhibits just density values around 1019 cm^(-2)
and mobility values clearly below 5000 cm2/Vs. The carrier density however depends much
on lithography and surface treatment after growth. Furthermore, the relaxation behavior and
critical thickness of HgTe grown on CdTe is determined and is in very good agreement with
theoretical prediction (d_c = 155 nm). The embedding of the HgTe bulk layer between HgCdTe
layers created a further huge improvement. Similar to the quantum well structures the carrier
mobility increases immensely while the carrier density levels at around 1x10^11 cm^(-2) at 0
V gate voltage as well. Additionally, the relaxation behavior and critical thickness of these
barrier layers has to be determined. HgCdTe grown on commercial CdTe shows a behavior as
predicted except the critical thickness which is slightly higher than expected (d_c = 850 nm).
Otherwise, the relaxation of HgCdTe grown on CdTe/GaAs:Si occurs in two parts. The layer
is fully strained up to 250 nm. Between 250 nm and 725 nm the HgCdTe film starts to relax
randomly up to 10 %. The relaxation behavior for thicknesses larger than 725 nm occurs than
linearly to the inverse layer thickness. A explanation is given due to rough interface conditions
and crystalline defects of the CdTe/GaAs:Si compared to the commercial CdTe substrate. HRXRD and AFM data support this statement. Another point is that the HgCdTe barriers protect the active HgTe layer and because of the high carrier mobilities the Hall measurements provide new transport data which have to be interpreted more in detail in the future. In addition, HgTe bulk samples show very interesting transport data by gating the sample from the top and the back. It is now possible to manipulate the carrier densities of the top and bottom surface states almost separately. The back gate consisting of the n-doped GaAs substrate and the thick insulating CdTe buffer can tune the carrier density for Delta(n) ~ 3x10^11 cm^(-2). This is sufficient to tune the Fermi energy from the p-type into the n-type region [138].
In this thesis it is shown that strained HgTe bulk layers exhibit superior transport data by embedding between HgCdTe barrier layers. The n-doped GaAs can here serve as a back gate.
Furthermore, MBE growth of high crystalline, undoped HgTe quantum wells shows also new
and extended transport output. Finally, it is notable that due to the investigated CdTe growth
on GaAs the Hg-based heterostructure MBE growth is partially independent from commercial
suppliers.
Neue physikalische Erkenntnisse vervollständigen die Sicht auf die Welt und erschließen gleichzeitig Wege für Folgeexperimente und technische Anwendungen. Das letzte Jahrzehnt der Festkörperforschung war vom zunehmenden Fokus der theoretischen und experimentellen Erkundung topologischer Materialien geprägt. Eine fundamentale Eigenschaft ist ihre Resistenz gegenüber solchen Störungen, welche spezielle physikalische Symmetrien nicht verletzen. Insbesondere die Topologischen Isolatoren - Halbleiter mit isolierenden Volumen- sowie gleichzeitig leitenden und spinpolarisierten Oberflächenzuständen - sind vielversprechende Kandidaten zur Realisierung breitgefächerter spintronischer Einsatzgebiete. Bis zur Verwirklichung von Quantencomputern und anderer, heute noch exotisch anmutender Konzepte bedarf es allerdings ein umfassenderes Verständnis der grundlegenden, physikalischen Zusammenhänge. Diese kommen vor allem an Grenzflächen zum Tragen, weshalb oberflächensensitive Methoden bei der Entdeckung der Topologischen Isolatoren eine wichtige Rolle spielten.
Im Rahmen dieser Arbeit werden daher strukturelle, elektronische und magnetische Eigenschaften Topologischer Isolatoren mittels Tieftemperatur-Rastertunnelmikroskopie und -spektroskopie sowie begleitenden Methoden untersucht.
Die Veränderung der Element-Ausgangskonzentration während dem Wachstum des prototypischen Topologischen Isolators Bi2Te3 führt zur Realisierung eines topologischen p-n Übergangs innerhalb des Kristalls. Bei einem spezifischen Verhältnis von Bi zu Te in der Schmelze kommt es aufgrund unterschiedlicher Erstarrungstemperaturen der Komponenten zu einer Ansammlung von Bi- und Te-reichen Gegenden an den gegenüberliegenden Enden des Kristalls. In diesen bildet sich infolge des jeweiligen Elementüberschusses durch Kristallersetzungen und -fehlstellen eine Dotierung des Materials aus. Daraus resultiert die Existenz eines Übergangsbereiches, welcher durch Transportmessungen verifiziert werden kann. Mit der räumlich auflösenden Rastertunnelmikroskopie wird diese Gegend lokalisiert und strukturell sowie elektronisch untersucht. Innerhalb des Übergangsbereiches treten charakteristische Kristalldefekte beider Arten auf - eine Defektunterdrückung bleibt folglich aus. Dennoch ist dort der Beitrag der Defekte zum Stromtransport aufgrund ihres gegensätzlichen Dotiercharakters vernachlässigbar, sodass der topologische Oberflächenzustand die maßgeblichen physikalischen Eigenschaften bestimmt. Darüber hinaus tritt der Übergangsbereich in energetischen und räumlichen Größenordnungen auf, die Anwendungen bei Raumtemperatur denkbar machen.
Neben der Veränderung Topologischer Isolatoren durch den gezielten Einsatz intrinsischer Kristalldefekte bieten magnetische Störungen die Möglichkeit zur Prüfung des topologischen Oberflächenzustandes auf dessen Widerstandsfähigkeit sowie der gegenseitigen Wechselwirkungen. Die Zeitumkehrinvarianz ist ursächlich für den topologischen Schutz des Oberflächenzustandes, weshalb magnetische Oberflächen- und Volumendotierung diese Symmetrie brechen und zu neuartigem Verhalten führen kann.
Die Oberflächendotierung Topologischer Isolatoren kann zu einer starken Bandverbiegung und einer energetischen Verschiebung des Fermi-Niveaus führen. Bei einer wohldosierten Menge der Adatome auf p-dotiertem Bi2Te3 kommt die Fermi-Energie innerhalb der Volumenzustands-Bandlücke zum Liegen. Folglich wird bei Energien rund um das Fermi-Niveau lediglich der topologische Oberflächenzustand bevölkert, welcher eine Wechselwirkung zwischen den Adatomen vermitteln kann. Für Mn-Adatome kann Rückstreuung beobachtet werden, die aufgrund der Zeitumkehrinvarianz in undotierten Topologischen Isolatoren verboten ist. Die überraschenderweise starken und fokussierten Streuintensitäten über mesoskopische Distanzen hinweg resultieren aus der ferromagnetischen Kopplung nahegelegener Adsorbate, was durch theoretische Berechnungen und Röntgendichroismus-Untersuchungen bestätigt wird. Gleichwohl wird für die Proben ein superparamagnetisches Verhalten beobachtet.
Im Gegensatz dazu führt die ausreichende Volumendotierung von Sb2Te3 mit V-Atomen zu einem weitreichend ferromagnetischen Verhalten. Erstaunlicherweise kann trotz der weitläufig verbreiteten Theorie Zeitumkehrinvarianz-gebrochener Dirac-Zustände und der experimentellen Entdeckung des Anormalen Quanten-Hall-Effektes in ähnlichen Probensystemen keinerlei Anzeichen einer spektroskopischen Bandlücke beobachtet werden. Dies ist eine direkte Auswirkung der dualen Natur der magnetischen Adatome: Während sie einerseits eine magnetisch induzierte Bandlücke öffnen, besetzen sie diese durch Störstellenresonanzen wieder. Ihr stark lokaler Charakter kann durch die Aufnahme ihrer räumlichen Verteilung aufgezeichnet werden und führt zu einer Mobilitäts-Bandlücke, deren Indizien durch vergleichende Untersuchungen an undotiertem und dotiertem Sb2Te3 bestätigt werden.
Two-dimensional (2D) topological insulators are a new class of materials with properties that are
promising for potential future applications in quantum computers. For example, stanene represents
a possible candidate for a topological insulator made of Sn atoms arranged in a hexagonal
lattice. However, it has a relatively fragile low-energy spectrum and sensitive topology. Therefore,
to experimentally realize stanene in the topologically non-trivial phase, a suitable substrate
that accommodates stanene without compromising these topological properties must be found.
A heterostructure consisting of a SiC substrate with a buffer layer of adsorbed group-III elements
constitutes a possible solution for this problem. In this work, 2D adatom systems of Al and In
were grown epitaxially on SiC(0001) and then investigated structurally and spectroscopically by
scanning tunneling microscopy (STM) and photoelectron spectroscopy.
Al films in the high coverage regime \( (\Theta_{ML}\approx2\) ML\( ) \) exhibit unusually large, triangular- and
rectangular-shaped surface unit cells. Here, the low-energy electron diffraction (LEED)
pattern is brought into accordance with the surface topography derived from STM. Another Al
reconstruction, the quasi-one-dimensional (1D) Al phase, exhibits a striped surface corrugation,
which could be the result of the strain imprinted by the overlayer-substrate lattice mismatch.
It is suggested that Al atoms in different surface areas can occupy hexagonal close-packed and
face-centered cubic lattice sites, respectively, which in turn lead to close-packed transition regions
forming the stripe-like corrugations. On the basis of the well-known herringbone reconstruction
from Au(111), a first structural model is proposed, which fits well to the structural data from
STM. Ultimately, however, thermal treatments of the sample could not generate lower coverage
phases, i.e. in particular, a buffer layer structure.
Strong metallic signatures are found for In high coverage films \( (\Theta_{ML}\approx3\) to \(2\) ML\() \) by
scanning tunneling spectroscopy (STS) and angle-resolved photoelectron spectroscopy (ARPES),
which form a \( (7\times7) \), \( (6\times4\sqrt{3}) \), and \( (4\sqrt{3}\times4\sqrt{3}) \) surface reconstruction. In all these In phases
electrons follow the nearly-free electron model. Similar to the Al films, thermal treatments could
not obtain the buffer layer system.
Surprisingly, in the course of this investigation a triangular In lattice featuring a \( (1\times1) \)
periodicity is observed to host massive Dirac-like bands at \( K/K^{\prime} \) in ARPES. Based on this
strong electronic similarity with graphene at the Brillouin zone boundary, this new structure is
referred to as \textit{indenene}. An extensive theoretical analysis uncovers the emergence of an electronic
honeycomb network based on triangularly arranged In \textit{p} orbitals. Due to strong atomic spin-orbit
coupling and a comparably small substrate-induced in-plane inversion symmetry breaking this
material system is rendered topologically non-trivial. In indenene, the topology is intimately
linked to a bulk observable, i.e., the energy-dependent charge accumulation sequence within the
surface unit cell, which is experimentally exploited in STS to confirm the non-trivial topological
character. The band gap at \( K/K^{\prime} \), a signature of massive Dirac fermions, is estimated by
ARPES to approximately 125 meV. Further investigations by X-ray standing wave, STM, and
LEED confirm the structural properties of indenene. Thus, this thesis presents the growth and
characterization of the novel quantum spin Hall insulator material indenene.
The presented thesis summarizes the results from four and a half years of intense lithography development on (Cd,Hg)Te/HgTe/(Cd,Hg)Te quantum well structures. The effort was motivated by the unique properties of this topological insulator. Previous work from Molenkamp at al.\ has proven that the transport through such a 2D TI is carried by electrons with opposite spin, counter-propagating in 1D channels along the sample edge. However, up to this thesis, the length of quantized spin Hall channels has never been reported to exceed 4 µm. Therefore, the main focus was put on a reproducible and easy-to-handle fabrication process that reveals the intrinsic material parameters.
Every single lithography step in macro as well as microscopic sample fabrication has been re-evaluated. In the Development, the process changes have been presented along SEM pictures, microgaphs and, whenever possible, measurement responses.
We have proven the conventional ion milling etch method to damage the remaining mesa and result in drastically lower electron mobilities in samples of microscopic size.
The novel KI:I2:HBr wet etch method for macro and microstructure mesa fabrication has been shown to leave the crystalline structure intact and result in unprecedented mobilities, as high as in macroscopic characterization Hall bars. Difficulties, such as an irregular etch start and slower etching of the conductive QW have been overcome by concentration, design and etch flow adaptations. In consideration of the diffusive regime, a frame around the EBL write field electrically decouples the structure mesa from the outside wafer. As the smallest structure, the frame is etched first and guarantees a non-different etching of the conductive layer during the redox reaction. A tube-pump method assures reproducible etch results with mesa heights below 300 nm. The PMMA etch mask is easy to strip and leaves a clean mesa with no redeposition. From the very first attempts, to the final etch process, the reader has been provided with the characteristics and design requirements necessary to enable the fabrication of nearly any mesa shape within an EBL write field of 200 µm.
Magneto resistance measurement of feed-back samples have been presented along the development chronology of wet etch method and subsequent lithography steps. With increasing feature quality, more and more physics has been revealed enabling detailed evaluation of smallest disturbances. The following lithography improvements have been implemented. They represent a tool-box for high quality macro and microstructure fabrication on (CdHg)Te/HgTe of almost any kind.
The optical positive resist ECI 3027 can be used as wet and as dry etch mask for structure sizes larger than 1 µm. It serves to etch mesa structures larger than the EBL write field.
The double layer PMMA is used for ohmic contact fabrication within the EBL write field. Its thickness allows to first dry etch the (Cd,Hg)Te cap layer and then evaporate the AuGe contact, in situ and self-aligned. Because of an undercut, up to 300 nm can be metalized without any sidewalls after the lift-off. An edge channel mismatch within the contact leads can be avoided, if the ohmic contacts are designed to reach close to the sample and beneath the later gate electrode.
The MIBK cleaning step prior to the gate application removes PMMA residuals and thereby improves gate and potential homogeneity.
The novel low HfO2-ALD process enables insulator growth into optical and EBL lift-off masks of any resolvable shape. Directly metalized after the insulator growth, the self-aligned method results in thin and homogeneous gate electrode reproducibly withholding gate voltages to +-10 V.
The optical negative resist ARN 4340 exhibits an undercut when developed. Usable as dry etch mask and lift-off resist, it enables an in-situ application of ohmic contacts first etching close to the QW, then metalizing AuGe. Up to 500 nm thickness, the undercut guarantees an a clean lift-off with no sidewalls.
The undertaken efforts have led to micro Hall bar measurements with Hall plateaus and SdH-oszillations in up to now unseen levels of detail.
The gap resistance of several micro Hall bars with a clear QSH signal have been presented in Quantum Spin Hall. The first to exhibit longitudinal resistances close to the expected h/2e2 since years, they reveal unprecedented details in features and characteristics. It has been shown that their protection against backscattering through time reversal symmetry is not as rigid as previously claimed. Values below and above 12.9 kΩ been explained, introducing backscattering within the Landauer-Büttiker formalism of edge channel transport. Possible reasons have been discussed. Kondo, interaction and Rashba-backscattering arising from density inhomogeneities close to the edge are most plausible to explain features on and deviations from a quantized value. Interaction, tunneling and dephasing mechanisms as well as puddle size, density of states and Rashba Fields are gate voltage dependent. Therefore, features in the QSH signal are fingerprints of the characteristic potential landscape.
Stable up to 11 K, two distinct but clear power laws have been found in the higher temperature dependence of the QSH in two samples. However, with ΔR = Tα, α = ¼ in one (QC0285) and α = 2 in the other (Q2745), none of the predicted dependencies could be confirmed. Whereas, the gap resistances of QC0285 remains QSH channel dominated up to 3.9 T and thereby confirmed the calculated lifting of the band inversion in magnetic field. The gate-dependent oscillating features in the QSH signal of Q2745 immediately increase in magnetic field. The distinct field dependencies allowed the assumption of two different dominant backscattering mechanisms.
Resulting in undisturbed magneto transport and unprecedented QSH measurements The Novel Micro Hall Bar Process has proven to enable the fabrication of a new generation of microstructures.
In this thesis, we investigate aspects of the physics of heavy-fermion systems and correlated topological insulators.
We numerically solve the interacting Hamiltonians that model the physical systems using quantum Monte Carlo algorithms
to access both ground-state and finite-temperature observables.
Initially, we focus on the metamagnetic transition in the Kondo lattice model for heavy fermions.
On the basis of the dynamical mean-field theory and the dynamical cluster approximation,
our calculations point towards a continuous transition, where the signatures of metamagnetism are linked to a Lifshitz transition of heavy-fermion bands.
In the second part of the thesis, we study various aspects of magnetic pi fluxes in the Kane-Mele-Hubbard model of a correlated topological insulator.
We describe a numerical measurement of the topological index, based on the localized mid-gap states that are provided by pi flux insertions.
Furthermore, we take advantage of the intrinsic spin degree of freedom of a pi flux to devise instances of interacting quantum spin systems.
In the third part of the thesis, we introduce and characterize the Kane-Mele-Hubbard model on the pi flux honeycomb lattice.
We place particular emphasis on the correlations effects along the one-dimensional boundary of the lattice and
compare results from a bosonization study with finite-size quantum Monte Carlo simulations.
Recently a new state of matter was discovered in which the bulk insulating state in a material is accompanied by conducting surface or edge states. This new state of matter can be distinguished from a conventional insulator phase by the topological properties of its band structure which led to the name "topological insulators". Experimentally, topological insulator states are mostly found in systems characterized by a band inversion compared to conventional systems. In most topological insulator systems, this is caused by a combination of energetically close bands and spin orbit coupling. Such properties are found in systems with heavy elements like Hg and Bi. And indeed, the first experimental discovery of a topological insulator succeeded in HgTe quantum wells and later also in BiSb bulk systems.
Topological insulators are of large interest due to their unique properties: In 2-dimensional topological insulators one dimensional edge states form without the need of an external magnetic field (in contrast to the quantum Hall effect). These edge states feature a linear band dispersion, a so called Dirac dispersion. The quantum spin Hall states are helical edge states, which means they consist of counterpropagating oppositely spin polarized edge channels. They are therefore of great potential for spintronic applications as well as building blocks for new more exotic states like Majorana Fermions. 3-dimensional topological insulators feature 2-dimensional surface states with only one Dirac band (also called Dirac cone) on each surface and an interesting spin texture where spin and momentum are locked perpendicular to each other in the surface plane. This unique surface band structure is predicted to be able to host several exotic states like e.g. Majorana Fermions (in combination with superconductors) and magnetic monopole like excitations.
This PhD thesis will summarize the discovery of topological insulators and highlights the developments on their experimental observations. The work focuses on HgTe which is up to now the only topological insulator material where the expected properties are unambiguously demonstrated in transport experiments. In HgTe, the topological insulator properties arise from the inversion of the Gamma_6 and Gamma_8 bands. The band inversion in HgTe is due to a combination of a high spin orbit splitting in Te and large energy corrections (due to the mass-velocity term) to the energy levels in Hg. Bulk HgTe, however, is a semimetal, which means for the conversion into a topological insulator a band gap has to be opened. In two dimensions (HgTe quantum well structures) this is achieved via quantum confinement, which opens a band gap between the quantum well subbands. In three dimensions, strain is used to lift the degeneracy of the semimetallic Gamma_8 bands opening up a band gap.
The thesis is structured as follows:
- The first chapter of this thesis will give a brief overview on discoveries in the field of topological insulators. It focuses on works relevant to experimental results presented in the following chapters. This includes a short outline of the early predictions and a summary of important results concerning 2-dimensional topological insulators while the final section discusses observations concerning 3-dimensional topological insulators.
- The discovery of the quantum spin Hall effect in HgTe marked the first experimental observation of a topological insulator. Chapter 2 will focus on HgTe quantum wells and the quantum spin Hall effect.
Above a critical thickness, HgTe quantum wells are predicted to host the quantum spin Hall state, the signature of a 2-dimensional topological insulator. HgTe quantum wells exhibiting low carrier concentrations and at the same time high carrier mobilities are required to be able to measure the quantum spin Hall effect. The growth of such high quality HgTe quantum wells was one of the major goals for this work. Continuous optimization of the substrate preparation and growth conditions resulted in controlled carrier densities down to a few 10^10 cm^-2. At the same time, carrier mobilities exceeding 1 x 10^6 cm^2/Vs have been achieved, which provides mean free paths of several micrometers in the material. Thus the first experimental evidence for the existence of the quantum spin Hall edge states succeeded in transport experiments on microstructures: When the Fermi energy was located in the bulk band gap a residual quantized resistance of 2e^2/h was found. Further experiments focused on investigating the nature of transport in this regime. By non-local measurements the edge state character could be established. The measured non-local resistances corresponded well with predictions from the Landauer-Büttiker theory applied to transport in helical edge channels.
In a final set of experiments the spin polarization of the edge channels was investigated. Here, we could make use of the advantage that HgTe quantum well structures exhibit a large Rashba spin orbit splitting. In systems with a large Rashba spin orbit splitting a spin accumulation is expected to occur at the edge of the sample perpendicular to a current flow. This so-called spin Hall effect was then used as a spin injector and detector. Using split gate devices it was possible to bring spin Hall and quantum spin Hall state into direct contact, which enabled an all electrical detection of the spin polarization of the quantum spin Hall edge channels.
- HgTe as a 3-dimensional topological insulator will be presented in chapter 3. Straining the HgTe layer enables the observation of topological insulator behavior. It was found that strain can be easily implemented during growth by using CdTe substrates. CdTe has a slightly larger lattice constant than HgTe and therefore leads to tensile strain in the HgTe layer as long as the growth is pseudomorphic. Magnetotransport studies showed the emergence of quantum Hall transport with characteristic signatures of a Dirac type bandstructure. Thus, this result marks the first observation of the quantum Hall effect in the surface states of a 3-dimensional topological insulator.
Transport experiments on samples fitted with a top gate enabled the identification of contributions from individual surfaces. Furthermore, the surface state quantum Hall effect was found to be surprisingly stable, perturbations due to additional bulk transport could not be found, even at high carrier densities of the system.
- Chapters 4 - 6 serve as in depth overviews of selected works: Chapter 4 presents a detailed overview on the all electrical detection of the spin Hall effect in HgTe quantum wells. The detection of the spin polarization of the quantum spin Hall effect is shown in chapter 5 and chapter 6 gives a detailed overview on the quantum Hall effect originating from the topological surface state in strained bulk HgTe.
The investigations discussed in this thesis pioneered the experimental work on the transport properties of topological insulator systems. The understanding of the fundamental properties of topological insulators enables new experiments in which e.g. the inclusion of magnetic dopants or the interplay between topological insulator and superconductors can be investigated in detail.
In this PhD thesis, the fingerprints of geometry and topology on low dimensional mesoscopic systems are investigated. In particular, holographic non-equilibrium transport properties of the quantum spin Hall phase, a two dimensional time reversal symmetric bulk insulating phase featuring one dimensional gapless helical edge modes are studied. In these metallic helical edge states, the spin and the direction of motion of the charge carriers are locked to each other and counter-propagating states at the same energy are conjugated by time reversal symmetry. This phenomenology entails a so called topological protection against elastic single particle backscattering by time reversal symmetry. We investigate the limitations of this topological protection by studying the influence of inelastic processes as induced by the interplay of phonons and extrinsic spin orbit interaction and by taking into account multi electron processes due to electron-electron interaction, respectively. Furthermore, we propose possible spintronics applications that rely on a spin charge duality that is uniquely associated with the quantum spin Hall phase. This duality is present in the composite system of two helical edge states with opposite helicity as realized on the two opposite edges of a quantum spin Hall sample with ribbon geometry. More conceptually speaking, the quantum spin Hall phase is the first experimentally realized example of a symmetry protected topological state of matter, a non-interacting insulating band structure which preserves an anti-unitary symmetry and is topologically distinct from a trivial insulator in the same symmetry class with totally localized and hence independent atomic orbitals. In the first part of this thesis, the reader is provided with a fairly self-contained introduction into the theoretical concepts underlying the timely research field of topological states of matter. In this context, the topological invariants characterizing these novel states are viewed as global analogues of the geometric phase associated with a cyclic adiabatic evolution. Whereas the detailed discussion of the topological invariants is necessary to gain deeper insight into the nature of the quantum spin Hall effect and related physical phenomena, the non-Abelian version of the local geometric phase is employed in a proposal for holonomic quantum computing with spin qubits in quantum dots.
This thesis examines the electronic properties of two materials that promise the realization and observation of novel exotic quantum phenomena. For this purpose, angle-resolved photoemission forms the experimental basis for the investigation of the electronic properties. Furthermore, the magnetic order is investigated utilizing X-ray dichroism measurements.
First, the bulk and surface electronic structure of epitaxially grown HgTe in its three-dimensional topological insulator phase is investigated. In this study, synchrotron radiation is used to address the three-dimensional band structure and orbital composition of the bulk states by employing photon-energy-dependent and polarization-dependent measurements, respectively. In addition, the topological surface state is examined on in situ grown samples using a laboratory photon source. The resulting data provide a means to experimentally localize the bulk band inversion in momentum space and to evidence the momentum-dependent change in the orbital character of the inverted bulk states.
Furthermore, a rather new series of van der Waals compounds, (MnBi\(_2\)Te\(_4\))(Bi\(_2\)Te\(_3\))\(_n\), is investigated. First, the magnetic properties of the first two members of the series, MnBi\(_2\)Te\(_4\) and MnBi\(_4\)Te\(_7\), are studied via X-ray absorption-based techniques. The topological surface state on the two terminations of MnBi\(_4\)Te\(_7\) is analyzed using circular dichroic, photon-energy-dependent, and spin-resolved photoemission. The topological state on the (MnBi\(_2\)Te\(_4\))-layer termination shows a free-standing Dirac cone with its Dirac point located in the bulk band gap. In contrast, on the (Bi\(_2\)Te\(_3\))-layer termination the surface state hybridizes with the bulk valences states, forming a spectral weight gap, and exhibits a Dirac point that is buried within the bulk continuum. Lastly, the lack of unambiguous evidence in the literature showing a temperature-dependent mass gap opening in these magnetic topological insulators is discussed through MnBi\(_2\)Te\(_4\).
Spin-Orbit Torques and Galvanomagnetic Effects Generated by the 3D Topological Insulator HgTe
(2021)
Nature shows us only the tail of the lion. But I have no doubt that the lion belongs with it even if he cannot reveal himself all at once. Albert Einstein
In my dissertation, I addressed the question of whether the 3D topological insulator mercury telluride (3D TI HgTe) is a suitable material for spintronics applications. This question was addressed by investigating the SOTs generated by the 3D TI HgTe in an adjacent ferromagnet (Permalloy) by using the ferromagnetic resonance technique (SOT-FMR).
In the first part of the dissertation, the reader was introduced to the mathematical description of the SOTs of a hybrid system consisting of a topological insulator (TI) and a ferromagnet (FM). Furthermore, the sample preparation and the measurement setup for the SOT-FMR measurements were discussed. Our SOT-FMR measurements showed that at low temperatures (T = 4.2 K) the out-of-plane component of the torque is dominant. At room temperature, both in-plane and out-of-plane components of the torque could be observed. From the symmetry of the mixing voltage (Figs. 3.14 and 3.15) we could conclude that the 3D TI HgTe may be efficient for the generation of spin torques in the permalloy [1]. The investigations reported here showed that the SOT efficiencies generated by the 3D TI HgTe are comparable with other existent topological insulators (see Fig. 3.17). We also discussed in detail the parasitic effects (such as thermovoltages) that can contribute to the correct interpretation of the spin torque efficiencies.
Although the results reported here provide several indications that the 3D TI HgTe might be efficient in exerting spin-torques in adjacent ferromagnets [2], the reader was repeatedly made aware that parasitic effects might contaminate the correct writing and reading of the information in the ferromagnet. These effects should be taken into consideration when interpreting results in the published literature claiming high spin-orbit torque efficiencies [2–4]. The drawbacks of the SOT-FMR measurement method led to a further development of our measurement concept, in which the ferromagnet on top of the 3D TI HgTe was replaced by a
spin-valve structure. In contrast with our measurements, in this measurement setup, the current flowing through the HgTe is known and changes in the spin-valve resistance can be read via the GMR effect.
Moreover, the SOT-FMR experiments required the application of an in-plane magnetic field up to 300 mT to define the magnetization direction in the ferromagnet. Motivated by this fact, we investigated the influence of an in-plane magnetic field in the magnetoresistance of the 3D TI HgTe. The surprising results of these measurements are described in the second part of the dissertation. Although the TI studied here is non-magnetic, its transversal MR (Rxy) showed an oscillating behavior that depended on the angle between the in-plane magnetic field and the electrical current. This effect is a typical property of ferromagnetic materials and is called planar Hall effect (PHE) [5, 6]. Moreover, it was also shown that the PHE amplitude (Rxy) and the longitudinal resistance (Rxx) oscillate as a function of the in-plane magnetic field amplitude for a wide range of carrier densities of the topological insulator.
The PHE was already described in another TI material (Bi2−xSbxTe3) [7]. The authors suggested as a possible mechanism the scattering of the electron off impurities that are polarized by an in-plane magnetic field. We critically discussed this and other theoretical proposed mechanisms existent in the literature [8, 9].
In this thesis, we attempted to explain the origin of the PHE in the 3D TI HgTe by anisotropies in the band structure of this material. The k.p calculations based on 6-orbitals were able to demonstrate that an interplay between Rashba, Dresselhaus, and in-plane magnetic field deforms the Fermi contours of the camel back band of the 3D TI HgTe, which could lead to anisotropies in its conductivity. However, the magnetic fields needed to experimentally observe this effect are as
high as 40 T, i.e., one order of magnitude higher than reported in our experiments. Additionally, calculations of the DoS to assess if there is a difference in the states for Bin parallel and Bin perpendicular to the current were, so far, inconclusive. Moreover, the complicated dependence of Rashba in the p-conducting
regime of HgTe [10] makes it not straightforward the inclusion of this term in the band structure calculations.
Despite the extensive efforts to understand the origin of the galvanomagnetic effects in the 3D TI HgTe, we could not determine a clear mechanism for the origin of the PHE and the MR oscillations studied in this thesis. However, our work clarifies and excludes a few mechanisms reported in the literature as the origin of these effects in the 3D TI HgTe. The major challenge, which still needs to be overcome, is to find a model that simultaneously explains the PHE, the gate dependence, and the oscillations in the magnetoresistance of the 3D TI HgTe as a function of the in-plane magnetic field.
To conclude, the author would like to express her hope to have brought the reader closer to the complexity of the questions addressed in this thesis and to have initiated them into the art of properly conducting electrical transport measurements on topological insulators with in-plane magnetic fields.
This thesis focuses on investigating magneto-transport properties of a ferromagnetic topological insulator (V,Bi,Sb)2Te3. This material is most famously known for exhibiting the quantum anomalous Hall effect, a novel quantum state of matter that has opened up possibilities for potential applications in quantum metrology as a quantum standard of resistance, as well as for academic investigations into unusual magnetic properties and axion electrodynamics. All of those aspects are investigated in the thesis.
The prediction and the experimental discovery of topological insulators has set the stage for a novel type of electronic devices. In contrast to conventional metals or semiconductors, this new class of materials exhibits peculiar transport properties at the sample surface, as conduction channels emerge at the topological boundaries of the system.
In specific materials with strong spin-orbit coupling, a particular form of a two-dimensional topological insulator, the quantum spin Hall state, can be observed.
Here, the respective one-dimensional edge channels are helical in nature, meaning that there is a locking of the spin orientation of an electron and its direction of motion.
Due to the symmetry of time-reversal, elastic backscattering off interspersed impurities is suppressed in such a helical system, and transport is approximately ballistic.
This allows in principle for the realization of novel energy-efficient devices, ``spintronic`` applications, or the formation of exotic bound states with non-Abelian statistics, which could be used for quantum computing.
The present work is concerned with the general transport properties of one-dimensional helical states. Beyond the topological protection mentioned above, inelastic backscattering can arise from various microscopic sources, of which the most prominent ones will be discussed in this Thesis. As it is characteristic for one-dimensional systems, the role of electron-electron interactions can be of major importance in this context.
First, we review well-established techniques of many-body physics in one dimension such as perturbative renormalization group analysis, (Abelian) bosonization, and Luttinger liquid theory. The latter allow us to treat electron interactions in an exact way.
Those methods then are employed to derive the corrections to the conductance in a helical transport channel, that arise from various types of perturbations.
Particularly, we focus on the interplay of Rashba spin-orbit coupling and electron interactions as a source of inelastic single-particle and two-particle backscattering. It is demonstrated, that microscopic details of the system, such as the existence of a momentum cutoff, that restricts the energy spectrum, or the presence of non-interacting leads attached to the system, can fundamentally alter the transport signature.
By comparison of the predicted corrections to the conductance to a transport experiment, one can gain insight about the microscopic processes and the structure of a quantum spin Hall sample.
Another important mechanism we analyze is backscattering induced by magnetic moments. Those findings provide an alternative interpretation of recent transport measurements in InAs/GaSb quantum wells.
This thesis deals with quantum Monte Carlo simulations of correlated low dimensional electron systems. The correlation that we have in mind is always given by the Hubbard type electron electron interaction in various settings. To facilitate this task, we develop the necessary methods in the first part. We develop the continuous time interaction expansion quantum algorithm in a manner suitable for the treatment of effective and non-equilibrium problems. In the second part of this thesis we consider various applications of the algorithms. First we examine a correlated one-dimensional chain of electrons that is subject to some form of quench dynamics where we suddenly switch off the Hubbard interaction. We find the light-cone-like Lieb-Robinson bounds and forms of restricted equilibration subject to the conserved quantities. Then we consider a Hubbard chain subject to Rashba spin-orbit coupling in thermal equilibrium. This system could very well be realized on a surface with the help of metallic adatoms. We find that we can analytically connect the given model to a model without spin-orbit coupling. This link enabled us to interpret various results for the standard Hubbard model, such as the single-particle spectra, now in the context of the Hubbard model with Rashba spin-orbit interaction. And finally we have considered a magnetic impurity in a host consisting of a topological insulator. We find that the impurity still exhibits the same features as known from the single impurity Anderson model. Additionally we study the effects of the impurity in the bath and we find that in the parameter regime where the Kondo singlet is formed the edge state of the topological insulator is rerouted around the impurity.
One of the most significant technological advances in history was driven by the utilization of a new material class: semiconductors.
Its most important application being the transistor, which is indispensable in our everyday life. The technological advance in the semiconductor industry, however, is about to slow down. Making transistors ever smaller to increase the performance and trying to reduce and deal with the dissipative heat will soon reach the limits dictated by quantum mechanics with Moore himself, predicting the death of his famous law in the next decade.
A possible successor for semiconductor transistors is the recently discovered material class of topological insulators. A material which in its bulk is insulating but has topological protected metallic surface states or edge states at its boundary. Their electrical transport characteristics include forbidden backscattering and spin-momentum-locking with the spin of the electron being perpendicular to its momentum. Topological insulators therefore offer an opportunity for high performance devices with low dissipation, and applications in spintronic where data is stored and processed at the same point.
The topological insulator Bi\(_2\)Se\(_3\) and related compounds offer relatively high energy band gaps and a rather simple band structure with a single dirac cone at the gamma point of the Brillouin zone. These characteritics make them ideal candidates to study the topological surface state in electrical transport experiments and explore its physics.
Novel appraches to the molecular beam epitaxy of core-shell nanowires in the group II telluride material system were explored in this work. Significant advances in growth spurred the development of a flexible and reliable platform for a charge transport characterization of the topological insulator HgTe in a tubular nanowire geometry. The transport results presented provide an important basis for the design of future studies that strive for the experimental realization of topological charge transport in the quantum wire limit.
Topological phenomena known from solid state physics have been transferred to a variety of other classical and quantum systems. Due to the equivalence of the Hamiltonian matrix describing tight binding models and the grounded circuit Laplacian describing an electrical circuit we can investigate such phenomena in circuits. By implementing different Hermitian topological models general suggestions on designing those types of circuit are worked out with the aim of minimizing unwanted coupling effects and parasitic admittances in the circuit. Here the existence and the spatial profile of topological states as well as the band structure of the model can be determined.
Due to the complex nature of electric admittance the investigations can be directly expanded to systems with broken Hermiticity. The particular advantages of the experimental investigation of non-exclusively topological phenomena by means of electric circuits come to light in the realization of non-Hermitian and non-linear models. Here we find limitation of the Hermitian bulk-boundary correspondence principle, purely real eigenvalues in non-Hermitian PT-symmetrical systems and edge localization of all eigenstates in non-Hermitian and non-reciprocal systems, which in literature is termed the non-Hermitian skin effect.
When systems obeying non-linear equations are studied, the grounded circuit Laplacian based on the Fourier-transform cannot be applied anymore. By combination of the connectivity of a topological system together with non-linear van der Pol oscillators self-activated and self-sustained topological edge oscillations can be found. These robust high frequency sinusoidal edge oscillations differ significantly from low frequency relaxation oscillations, which can be found in the bulk of the system.
Einerseits besteht die einfachste Möglichkeit zum Ladungs- und Informationstransport zwischen zwei Punkten in deren direkter Verbindung durch eindimensionale Kanäle. Andererseits besitzen topologische Materialien exotische und äußerst vorteilhafte Eigenschaften, weshalb es nahe liegt, dass schon bald neue Anwendungen aus ihnen realisiert werden. Wenn diese beiden Entwicklungen zusammenkommen, dann ist ein grundlegendes Verständnis von Quanteninterferenz oder Hybridisierungseffekten in eindimensionalen, topologischen Kanälen von fundamentaler Wichtigkeit. Deshalb werden in der vorliegenden Arbeit Wechselwirkungen von eindimensionalen, topologisch geschützten Kantenzuständen, die an ungeradzahligen Stufenkanten auf der (001)–Oberfläche von Pb1−xSnxSe auftreten, untersucht. Aufgrund der lateralen Lokalisierung auf wenige Nanometer um eine Stufenkante herum und der Notwendigkeit zwischen gerad- und ungeradzahligen Stufenkantenhöhen zu unterscheiden, bieten sich die Rastertunnelmikroskopie und -spektroskopie als Methoden an. Die neu entdeckten Kopplungs- bzw. Wechselwirkungseffekte zwischen benachbarten Kantenzuständen treten auf, sobald der Stufe zu Stufe Abstand einen kritischen Wert von dkri ≈ 25nm unterschreitet. Dieses Kriterium kann durch verschiedene räumliche Anordnungen von Stufenkanten erfüllt werden. Infolgedessen werden sich kreuzende, parallel verlaufende und zusammenlaufende Stufenkanten genauer untersucht. Bei letzteren verändert sich entlang der Struktur kontinuierlich der Abstand und damit die Kopplungsstärke zwischen den beiden Randkanälen. Infolgedessen wurden drei Koppelungsregime identifiziert. (I) Ausgehend von einer schwachen Wechselwirkung zeigt der für die Kantenzustände charakteristische Peak im Spektrum zunächst eine Verbreiterung und Verminderung der Intensität. (II) Mit weiter zunehmender Wechselwirkung beginnt sich der Zustand in zwei Peaks aufzuspalten, sodass ab dkri ≈ 15nm an beiden Stufenkanten durchgehen eine Doppelpeak zu beobachten ist . Mit weiter abnehmendem Abstand erreicht die Aufspaltung Werte von einigen 10 meV, während sich die Intensität weiter reduziert. (III) Sobald zwei Stufenkanten weniger als etwa 5nm voneinander getrennt sind, konvergieren aufgrund der schwindenden Intensität und des sinkenden energetischen Abstands der beiden Peaks zu den van Hove Singularitäten die Spektren an den Stufenkanten gegen das Spektrum über einer Terrasse. i Die Aufspaltung verläuft in den Bereichen I und II asymmetrisch, d. h. ein Peak verbleibt ungefähr bei der Ausgangsenergie, während der andere mit zunehmender Kopplung immer weiter weg schiebt. Bezüglich der Asymmetrie kann kein Unterschied festgestellt werden, ob die zusammenlaufenden Stufenkanten eine Insel oder Fehlstelleninsel bilden oder ob die Stufenkanten sogar gänzlich parallel verlaufen. Es zeigt sich keine Präferenz, ob zunächst der niederenergetische oder der hochenergetische Peak schiebt. Erst im Regime starker Kopplung (III) kann beobachtet werden, dass beide Peaks die Ausgangsenergie deutlich verlassen. Im Gegensatz dazu kann bei sich kreuzenden Stufen ein erheblicher Einfluss der Geometrie, in Form des eingeschlossenen Winkels, auf das Spektrum beobachtet werden. Unabhängig vom Winkel existiert am Kreuzungspunkt selbst kein Kantenzustand mehr. Die Zustände an den vier Stufen beginnen, abhängig vom Winkel, etwa 10-15nm vor dem Kreuzungspunkt abzuklingen. Überraschenderweise zeigt sich dabei, dass im Fall rechtwinkliger Stufen gar keine Aufspaltung zu beobachten ist, während bei allen anderen Winkeln ein Doppelpeak festgestellt werden kann. Diese Entdeckung deutet auf Orthogonalität bezüglich einer Quantenzahl bei den beteiligten Kantenzustände hin. Neben einer nur theoretisch vorhergesagten Spinpolarisation kann dieser Effekt auch von dem orbitalem Charakter der beteiligten Dirac–Kegel verursacht sein. Da der topologische Schutz in Pb1−xSnxSe durch Kristallsymmetrien garantiert ist, wird als letzter intrinsischer Effekt der Einfluss von eindimensionalen Defekten auf die Kantenzustände untersucht. Berücksichtigt werden dabei ein nicht näher klassifizierbarer, oberflächennaher Defekt und Schraubversetzungen. In beiden Fällen kann ebenfalls eine Aufspaltung des Kantenzustands in einen Doppelpeak gezeigt werden. Im zweiten Teil dieser Arbeit werden die Grundlagen für eine Wiederverwendung von (Pb,Sn)Se–Oberflächen bei zukünftige Experimenten mit (magnetischen) Adatomen geschaffen. Durch Kombination von Inoenzerstäubung und Tempern wird dabei nicht nur eine gereinigte Oberfläche erzeugt, sondern es kann auch das Ferminiveau gezielt erhöht oder gesenkt werden. Dieser Effekt beruht auf eine Modifikation der Sn– Konzentration und der von ihr kontrollierten Anzahl an Defektelektronen. Als letztes sind erste Messungen an Cu- und Fe–dotierte Proben gezeigt. Durch die Adatome tritt eine n–Dotierung auf, welche den Dirac–Punkt des Systems in Richtung des Ferminiveaus verschiebt. Sobald er dieses erreicht hat kommt es zu Wechselwirkungsphänomenen an freistehenden Stufenkanten. Dies führt zu einer Doppelpeakstruktur mit einer feinen Aufspaltung von wenigen meV. Das Phänomen ist auf ein schmales Energiefenster beschränkt, bei dem die Lage des Dirac–Punkts nur etwa 5 meV (in beide Richtungen) von der des Ferminiveaus abweichen darf.
The topic of this PhD thesis is the combination of topologically non-trivial phases with correlation effects stemming from Coulomb interaction between the electrons in a condensed matter system. Emphasis is put on both emerging benefits as well as hindrances, e.g. concerning the topological protection in the presence of strong interactions.
The physics related to topological effects is established in Sec. 2. Based on the topological band theory, we introduce topological materials including Chern insulators, topological insulators in two and three dimensions as well as Weyl semimetals. Formalisms for a controlled treatment of Coulomb correlations are presented in Sec. 3, starting with the topological field theory. The Random Phase Approximation is introduced as a perturbative approach, while in the strongly interacting limit the theory of quantum Hall ferromagnetism applies. Interactions in one dimension are special, and are treated through the Luttinger liquid description. The section ends with an overview of the expected benefits offered by the combination of topology and interactions, see Sec. 3.3.
These ideas are then elaborated in the research part. In Chap. II, we consider weakly interacting 2D topological insulators, described by the Bernevig-Hughes-Zhang model. This is applicable, e.g., to quantum well structures made of HgTe/CdTe or InAs/GaSb. The bulk band structure is here a mixture stemming from linear Dirac and quadratic Schrödinger fermions. We study the low-energy excitations in Random Phase Approximation, where a new interband plasmon emerges due to the combined Dirac and Schrödinger physics, which is absent in the separate limits. Already present in the undoped limit, one finds it also at finite doping, where it competes with the usual intraband plasmon. The broken particle-hole symmetry in HgTe quantum wells allows for an effective separation of the two in the excitation spectrum for experimentally accessible parameters, in the right range for Raman or electron loss spectroscopy. The interacting bulk excitation spectrum shows here clear differences between the topologically trivial and topologically non-trivial regime. An even stronger signal in experiments is expected from the optical conductivity of the system. It thus offers a quantitative way to identify the topological phase of 2D topological insulators from a bulk measurement.
In Chap. III, we study a strongly interacting system, forming an ordered, quantum Hall ferromagnetic state. The latter can arise also in weakly interacting materials with an applied strong magnetic field. Here, electrons form flat Landau levels, quenching the kinetic energy such that Coulomb interaction can be dominant. These systems define the class of quantum Hall topological insulators: topologically non-trivial states at finite magnetic field, where the counter-propagating edge states are protected by a symmetry (spatial or spin) other than time-reversal. Possible material realizations are 2D topological insulators like HgTe heterostructures and graphene. In our analysis, we focus on the vicinity of the topological phase transition, where the system is in a strongly interacting quantum Hall ferromagnetic state. The bulk and edge physics can be described by a nonlinear \sigma-model for the collective order parameter of the ordered state. We find that an emerging, continuous U(1) symmetry offers topological protection. If this U(1) symmetry is preserved, the topologically non-trivial phase persists in the presence of interactions, and we find a helical Luttinger liquid at the edge. The latter is highly tunable by the magnetic field, where the effective interaction strength varies from weakly interacting at zero field, K \approx 1, to diverging interaction strength at the phase transition, K -> 0.
In the last Chap. IV, we investigate whether a Weyl semimetal and a 3D topological insulator phase can exist together at the same time, with a combined, hybrid surface state at the joint boundaries. An overlap between the two can be realized by Coulomb interaction or a spatial band overlap of the two systems. A tunnel coupling approach allows us to derive the hybrid surface state Hamiltonian analytically, enabling a detailed study of its dispersion relation. For spin-symmetric coupling, new Dirac nodes emerge out of the combination of a single Dirac node and a Fermi arc. Breaking the spin symmetry through the coupling, the dispersion relation is gapped and the former Dirac node gets spin-polarized. We propose experimental realizations of the hybrid physics, including compressively strained HgTe as well as heterostructures of topological insulator and Weyl semimetal materials, connected to each other, e.g., by Coulomb interaction.
A novel growth method has been developed, allowing for the growth of strained HgTe shells on CdTe nanowires (NWs). The growth of CdTe-HgTe core-shell NWs required high attention in controlling basic parameters like substrate temperature and the intensity of supplied material fluxes. The difficulties in finding optimized growth conditions have been successfully overcome in this work.
We found the lateral redistribution of liquid growth seeds with a ZnTe growth start to be crucial to trigger vertical CdTe NW growth. Single crystalline zinc blende CdTe NWs grew, oriented along [111]B. The substrate temperature was the most critical parameter to achieve straight and long wires. In order to adjust it, the growth was monitored by reflection high-energy electron diffraction, which was used for fine tuning of the temperature over time in each growth run individually. For optimized growth conditions, a periodic diffraction pattern allowed for the detailed analysis of atomic arrangement on the surfaces and in the bulk. The ability to do so reflected the high crystal quality and ensemble uniformity of our CdTe NWs. The NW sides were formed by twelve stable, low-index crystalline facets. We observed two types stepped and polar sides, separated by in total six flat and non-polar facets.
The high crystalline quality of the cores allowed to grow epitaxial HgTe shells around. We reported on two different heterostructure geometries. In the first one, the CdTe NWs exhibit a closed HgTe shell, while for the second one, the CdTe NWs are overgrown mainly on one side. Scanning electron microscopy and scanning transmission electron microscopy confirmed, that many of the core-shell NWs are single crystalline zinc blende and have a high uniformity. The symmetry of the zinc blende unit cell was reduced by residual lattice strain. We used high-resolution X-ray diffraction to reveal the strain level caused by the small lattice mismatch in the heterostructures. Shear strain has been induced by the stepped hetero-interface, thereby stretching the lattice of the HgTe shell by 0.06 % along a direction oriented with an angle of 35 ° to the interface.
The different heterostructures obtained, were the base for further investigation of quasi-one-dimensional crystallites of HgTe. We therefore developed methods to reliably manipulate, align, localize and contact individual NWs, in order to characterize the charge transport in our samples. Bare CdTe cores were insulating, while the HgTe shells were conducting. At low temperature we found the mean free path of charge carriers to be smaller, but the phase coherence length to be larger than the sample size of several hundred nanometers. We observed universal conductance fluctuations and therefore drew the conclusion, that the trajectories of charge carriers are defined by elastic backscattering at randomly distributed scattering sites. When contacted with superconducting leads, we saw induced superconductivity, multiple Andreev reflections and the associated excess current. Thus, we achieved HgTe/superconductor interfaces with high interfacial transparency.
In addition, we reported on the appearance of peaks in differential resistance at Delta/e for HgTe-NW/superconductor and 2*Delta/e for superconductor/HgTe-NW/superconductor junctions, which is possibly related to unconventional pairing at the HgTe/superconductor interface. We noticed that the great advantage of our self-organized growth is the possibility to employ the metallic droplet, formerly seeding the NW growth, as a superconducting contact. The insulating wire cores with a metallic droplet at the tip have been overgrown with HgTe in a fully in-situ process. A very high interface quality was achieved in this case.
Topologische Isolatoren gehören zu einer Klasse von Materialien, an deren Realisation im Rahmen der zweiten quantenmechanischen Revolution gearbeitet wird. Einerseits sind zahlreiche Fragestellungen zu diesen Materialen und deren Nutzbarmachung noch nicht beantwortet, andererseits treiben vielversprechende Anwendungen im Feld der Quantencomputer und Spintronik die Lösung dieser Fragen voran. Topologische Rand- bzw. Oberflächenzustände wurden für unterschiedlichste Materialien und Strukturen theoretisch vorhergesagt, so auch für GaSb/InAs Doppelquantenfilme und Bi2Se3. Trotz intensiver Forschungsarbeiten und großer Fortschritte bedürfen viele Prozesse v. a. im Bereich der Probenherstellung und Verarbeitung noch der Optimierung. Die vorliegende Arbeit präsentiert Ergebnisse zur Molekularstahlepitaxie, zur Probenfertigung sowie zu elektro-optisch modulierter Transportuntersuchung von GaSb/InAs Doppelquantenfilmen und der epitaktischen Fertigung von Bi2Se3 Nanostrukturen.
Im ersten Teil dieser Arbeit werden die Parameter zur Molekularstrahlepitaxie sowie die Anpassung der Probenfertigung von GaSb/InAs Doppelquantenfilmen an material- und untersuchungsbedingte Notwendigkeiten beschrieben. Dieser verbesserte Prozess ermöglicht die Fertigung quantitativ vergleichbarer Probenserien. Anschließend werden Ergebnisse für Strukturen mit variabler InAs Schichtdicke unter elektrostatischer Kontrolle mit einem Frontgate präsentiert. Auch mit verbessertem Prozess zeigten sich Leckströme zum Substrat. Diese erschweren eine elektrostatische Kontrolle über Backgates. Die erstmals durch optische Anregung präsentierte Manipulation der Ladungsträgerart sowie des Phasenzustandes in GaSb/InAs Doppelquantenfilmen bietet eine Alternative zu problembehafteten elektrostatisch betriebenen Gates.
Im zweiten Teil wird die epitaktische Herstellung von Bi2Se3 Nanostrukturen gezeigt. Mit dem Ziel, Vorteile aus dem erhöhten Oberfläche-zu-Volumen Verhältnis zu ziehen, wurden im Rahmen dieser Arbeit erstmals Bi2Se3 Nanodrähte und -flocken mittels Molekularstrahlepitaxie für die Verwendung als topologischer Isolator hergestellt.
Ein Quantensprung – Kapitel 1 führt über die umgangssprachliche Wortbedeutung des Quantensprungs und des damit verbundenen Modells der Quantenmechanik in das Thema. Die Anwendung dieses Modells auf Quanten-Ensembles und dessen technische Realisation wird heute als erste Quantenmechanische Revolution bezeichnet und ist aus unserem Alltag nicht mehr wegzudenken. Im Rahmen der zweiten Quantenmechanischen Revolution soll nun die Anwendung auf einzelne Zustände realisiert und technisch nutzbar gemacht werden. Hierbei sind topologische Isolatoren ein vielversprechender Baustein. Es werden das Konzept des topologischen Isolators sowie die Eigenschaften der beiden in dieser Arbeit betrachteten Systeme beschrieben: GaSb/InAs Doppelquantenfilme und Bi2Se3 Nanostrukturen.
GaSb/InAs Doppelquantenfilme
Kapitel 2 beschreibt die notwendigen physikalischen und technischen Grundlagen. Ausgehend von der Entdeckung des Hall-Effekts 1879 werden die Quanten-Hall-Effekte eingeführt. Quanten-Spin-Hall-Isolatoren oder allgemeiner topologische Isolatoren sind Materialien mit einem isolierenden Inneren, weisen an der Oberfläche aber topologisch geschützte Zustände auf. Doppelquantenfilme aus GaSb/InAs, die in AlSb gebettet werden, weisen – abhängig vom Aufbau der Heterostruktur – eine typische invertierte Bandstruktur auf und sind ein vielversprechender Kandidat für die Nutzbarmachung der topologischen Isolatoren. GaSb, InAs und AlSb gehören zur 6,1 Ångström-Familie, welche für ihre opto-elektronischen Eigenschaften bekannt ist und häufig verwendet wird. Die Eigenschaften sowie die technologischen Grundlagen der epitaktischen Fertigung von Heterostrukturen aus den Materialien der 6,1 Ångström-Familie mittels Molekularstrahlepitaxie werden besprochen. Abschließend folgen die Charakterisierungs- und Messmethoden. Ein Überblick über die Literatur zu GaSb/InAs Doppelquantenfilmen in Bezug auf topologische Isolatoren rundet dieses Kapitel ab.
Zu Beginn dieser Arbeit stellten Kurzschlusskanäle eine Herausforderung für die Detektion der topologischen Randkanäle dar. Kapitel 3 behandelt Lösungsansätze hierfür und beschreibt die Verbesserung der Herstellung von GaSb/InAs Doppelquantenfilm-Strukturen mit Blick auf die zukünftige Realisation topologischer Randkanäle. In Abschnitt 3.1 werden numerische Simulationen präsentiert, die sich mit der Inversion der elektronischen Niveaus in Abhängigkeit der GaSb und InAs Schichtdicken dGaSb und dInAs beschäftigen. Ein geeigneter Schichtaufbau für Strukturen mit invertierter Bandordnung liegt im Parameterraum von 8 nm ≾ dInAs ≾ 12 nm und 8 nm ≾ dGaSb ≾ 10 nm. Abschnitt 3.2 beschreibt die epitaktische Herstellung von GaSb/InAs Doppelquantenfilmen mittels Molekularstrahlepitaxie. Die Fertigung eines GaSb Quasisubstrats auf ein GaAs Substrat wird präsentiert und anschließend der Wechsel auf native GaSb Substrate mit einer reduzierten Defektdichte sowie reproduzierbar hoher Probenqualität begründet. Ein Wechseln von binärem AlSb auf gitterangepasstes AlAsSb erlaubt die Verwendung dickerer Barrieren. Versuche, eine hinlängliche Isolation des Backgates durch das Einbringen einer dickeren unteren Barriere zu erreichen, werden in diesem Abschnitt diskutiert. In Abschnitt 3.3 wird die Optimierung der Probenprozessierung gezeigt. Die Kombination zweier angepasster Ätzprozesse – eines trockenchemischen und eines sukzessive folgenden nasschemischen Schrittes – liefert zusammen mit der Entfernung von Oberflächenoxiden reproduzierbar gute Ergebnisse. Ein materialselektiver Ätzprozess mit darauffolgender direkter Kontaktierung des InAs Quantenfilmes liefert gute Kontaktwiderstände, ohne Kurzschlusskanäle zu erzeugen. Abschnitt 3.4 gibt einen kompakten Überblick, über den im weiteren Verlauf der Arbeit verwendeten „best practice“ Prozess.
Mit diesem verbesserten Prozess wurden Proben mit variabler InAs Schichtdicke gefertigt und bei 4,2 K auf ihre Transporteigenschaften hin untersucht. Dies ist in Kapitel 4 präsentiert und diskutiert. Abschnitt 4.1 beschreibt die Serie aus drei Proben mit GaSb/InAs Doppelquantenfilm in AlSb Matrix mit einer variablen InAs Schichtdicke. Die InAs Schichtdicke wurde über numerische Simulationen so gewählt, dass je eine Probe im trivialen Regime, eine im invertierten Regime und eine am Übergang liegt. Gezeigt werden in Kapitel 4.2 Magnetotransportmessungen für konstante Frontgatespannungen sowie Messungen mit konstantem Magnetfeld gegen die Frontgatespannung. Die Messungen bestätigen eine Fertigung quantitativ vergleichbarer Proben, zeigen aber auch, dass keine der Proben im topologischen Regime liegt. Hierfür kommen mehrere Ursachen in Betracht: Eine Überschätzung der Hybridisierung durch die numerische Simulation, zu geringe InAs Schichtdicken in der Fertigung oder ein asymmetrisches Verschieben mit nur einem Gate (Kapitel 4.3). Zur Reduktion der Volumenleitfähigkeit wurden Al-haltigen Schichten am GaSb/InAs Übergang eingebracht. Die erwartete Widerstandssteigerung konnte in ersten Versuchen nicht gezeigt werde.
Die in Kapitel 5 gezeigte optische Manipulation des dominanten Ladungsträgertyps der InAs/GaSb-Doppelquantentöpfe gibt eine zusätzliche Kontrollmöglichkeit im Phasendiagramm. Optische Anregung ermöglicht den Wechsel der Majoritätsladungsträger von Elektronen zu Löchern. Dabei wird ein Regime durchlaufen, in dem beide Ladungsträger koexistieren. Dies weist stark auf eine Elektron-Loch-Hybridisierung mit nichttrivialer topologischer Phase hin. Dabei spielen zwei unterschiedliche physikalische Prozesse eine Rolle, die analog eines Frontgates bzw. eines Backgates wirken. Der Frontgate Effekt beruht auf der negativ persistenten Photoleitfähigkeit, der Backgate Effekt fußt auf der Akkumulation von Elektronen auf der Substratseite. Das hier gezeigte optisch kontrollierte Verschieben der Zustände belegt die Realisation von opto-elektronischem Schalten zwischen unterschiedlichen topologischen Phasen. Dies zeigt die Möglichkeit einer optischen Kontrolle des Phasendiagramms der topologischen Zustände in GaSb/InAs Doppelquantenfilmen. In Abschnitt 5.1 wird die optische Verstimmung von GaSb/InAs Quantenfilmen gezeigt und erklärt. Sie wird in Abhängigkeit von der Temperatur, der Anregungswellenlänge sowie der Anregungsintensität untersucht. Kontrollversuche an Proben mit einem unterschiedlichen Strukturaufbau zeigen, dass das Vorhandensein eines Übergitters auf der Substratseite der Quantenfilmstruktur essentiell für die Entstehung der Backgate-Wirkung ist (Abschnitt 5.2). Abschließend werden in Abschnitt 5.3 die Erkenntnisse zur optischen Kontrolle zusammengefasst und deren Möglichkeiten, wie optisch definierte topologischen Phasen-Grenzflächen, diskutiert.
Bi2Se3 Nanostrukturen
Mit Blick auf die Vorteile eines erhöhten Oberfläche-zu-Volumen Verhältnisses ist die Verwendung von Nanostrukturen für das Anwendungsgebiet der dreidimensionalen topologischen Isolatoren effizient. Mit dem Ziel, diesen Effekt für die Realisation des topologischen Isolators in Bi2Se3 auszunutzen, wurde im Rahmen dieser Arbeit erstmalig das Wachstum von Bi2Se3 Nanodrähten und -flocken mit Molekularstrahlepitaxie realisiert. In Kapitel 6 werden technische und physikalische Grundlagen hierzu erläutert (Abschnitt 6.1). Ausgehend von einer Einführung in dreidimensionale topologische Isolatoren werden die Eigenschaften des topologischen Zustandes in Bi2Se3 gezeigt. Darauf folgen die Kristalleigenschaften von Bi2Se3 sowie die Erklärung des epitaktischen Wachstums von Nanostrukturen mit Molekularstrahlepitaxie. In Abschnitt 6.2 schließt sich die Beschreibung der epitaktischen Herstellung an. Die Kristallstruktur wurde mittels hochauflösender Röntgendiffraktometrie und Transmissionselektronenmikroskopie als Bi2Se3 identifiziert. Rasterelektronenmikroskopie-Aufnahmen zeigen Nanodrähte und Nanoflocken auf mit Gold vorbehandelten bzw. nicht mit Gold vorbehandelten Proben. Der Wachstumsmechanismus für Nanodrähte kann nicht zweifelsfrei definiert werden. Das Fehlen von Goldtröpfchen an der Drahtspitze legt einen wurzelbasierten Wachstumsmechanismus nahe (Abschnitt 6.3).